arXiv subjects

Sebastien Roch

Publications and source records attributed to Sebastien Roch.

At least 37 records · Page 2Linked to original sources

arXiv2017

Species tree estimation using ASTRAL: how many genes are enough?

Species tree reconstruction from genomic data is increasingly performed using methods that account for sources of gene tree discordance such as incomplete lineage sorting. One popular method for reconstructing species trees from unrooted gene tree topologies is ASTRAL. In this paper, we derive theoretical sample complexity results for the number of genes required by ASTRAL to guarantee reconstruction of the correct species tree with high probability. We also validate those theoretical bounds in a simulation study. Our results indicate that ASTRAL requires $\mathcal{O}(f^{-2} \log n)$ gene trees to reconstruct the species tree correctly with high probability where n is the number of species and f is the length of the shortest branch in the species tree. Our simulations, which are the first to test ASTRAL explicitly under the anomaly zone, show trends consistent with the theoretical bounds and also provide some practical insights on the conditions where ASTRAL works well.

q-bio.PE↗

arXiv2017

Generalized least squares can overcome the critical threshold in respondent-driven sampling

In order to sample marginalized and/or hard-to-reach populations, respondent-driven sampling (RDS) and similar techniques reach their participants via peer referral. Under a Markov model for RDS, previous research has shown that if the typical participant refers too many contacts, then the variance of common estimators does not decay like $O(n^{-1})$, where $n$ is the sample size. This implies that confidence intervals will be far wider than under a typical sampling design. Here we show that generalized least squares (GLS) can effectively reduce the variance of RDS estimates. In particular, a theoretical analysis indicates that the variance of the GLS estimator is $O(n^{-1})$. We then derive two classes of feasible GLS estimators. The first class is based upon a Degree Corrected Stochastic Blockmodel for the underlying social network. The second class is based upon a rank-two model. It might be of independent interest that in both model classes, the theoretical results show that it is possible to estimate the spectral properties of the population network from the sampled observations. Simulations on empirical social networks show that the feasible GLS (fGLS) estimators can have drastically smaller error and rarely increase the error. A diagnostic plot helps to identify where fGLS will aid estimation. The fGLS estimators continue to outperform standard estimators even when they are built from a misspecified model and when there is preferential recruitment.

math.ST↗

arXiv2017

Distance-based species tree estimation: information-theoretic trade-off between number of loci and sequence length under the coalescent

We consider the reconstruction of a phylogeny from multiple genes under the multispecies coalescent. We establish a connection with the sparse signal detection problem, where one seeks to distinguish between a distribution and a mixture of the distribution and a sparse signal. Using this connection, we derive an information-theoretic trade-off between the number of genes, $m$, needed for an accurate reconstruction and the sequence length, $k$, of the genes. Specifically, we show that to detect a branch of length $f$, one needs $m = Θ(1/[f^{2} \sqrt{k}])$.

math.PR↗

arXiv2017

Species Trees are Recoverable from Unrooted Gene Tree Topologies Under a Constant Rate of Horizontal Gene Transfer

Reconstructing the tree of life from molecular sequences is a fundamental problem in computational biology. Modern data sets often contain a large number of genes, which can complicate the reconstruction problem due to the fact that different genes may undergo different evolutionary histories. This is the case in particular in the presence of horizontal genetic transfer (HGT), where a gene is inherited from a distant species rather than an immediate ancestor. Such an event produces a gene tree which is distinct from, but related to, the species phylogeny. In previous work, a natural stochastic models of HGT was introduced and studied. It was shown, both in simulation and theoretical studies, that a species phylogeny can be reconstructed from gene trees despite surprisingly high rates of HGT under this model. Rigorous lower and upper bounds on this achievable rate were also obtained, but a large gap remained. Here we close this gap, up to a constant. Specifically we show that a species phylogeny can be reconstructed correctly from gene trees even when, on each gene, each edge of the species tree has a constant probability of being the location of an HGT event. Our new reconstruction algorithm, which relies only on unrooted gene tree topologies, builds the tree recursively from the leaves and runs in polynomial time. We also provide a matching bound in the negative direction (up to a constant) and extend our results to some cases where gene trees are not perfectly known.

math.PR↗

arXiv2017

Phase transition in the sample complexity of likelihood-based phylogeny inference

Reconstructing evolutionary trees from molecular sequence data is a fundamental problem in computational biology. Stochastic models of sequence evolution are closely related to spin systems that have been extensively studied in statistical physics and that connection has led to important insights on the theoretical properties of phylogenetic reconstruction algorithms as well as the development of new inference methods. Here, we study maximum likelihood, a classical statistical technique which is perhaps the most widely used in phylogenetic practice because of its superior empirical accuracy. At the theoretical level, except for its consistency, that is, the guarantee of eventual correct reconstruction as the size of the input data grows, much remains to be understood about the statistical properties of maximum likelihood in this context. In particular, the best bounds on the sample complexity or sequence-length requirement of maximum likelihood, that is, the amount of data required for correct reconstruction, are exponential in the number, $n$, of tips---far from known lower bounds based on information-theoretic arguments. Here we close the gap by proving a new upper bound on the sequence-length requirement of maximum likelihood that matches up to constants the known lower bound for some standard models of evolution. More specifically, for the $r$-state symmetric model of sequence evolution on a binary phylogeny with bounded edge lengths, we show that the sequence-length requirement behaves logarithmically in $n$ when the expected amount of mutation per edge is below what is known as the Kesten-Stigum threshold. In general, the sequence-length requirement is polynomial in $n$. Our results imply moreover that the maximum likelihood estimator can be computed efficiently on randomly generated data provided sequences are as above.

math.PR↗

arXiv2017

Circular Networks from Distorted Metrics

Trees have long been used as a graphical representation of species relationships. However complex evolutionary events, such as genetic reassortments or hybrid speciations which occur commonly in viruses, bacteria and plants, do not fit into this elementary framework. Alternatively, various network representations have been developed. Circular networks are a natural generalization of leaf-labeled trees interpreted as split systems, that is, collections of bipartitions over leaf labels corresponding to current species. Although such networks do not explicitly model specific evolutionary events of interest, their straightforward visualization and fast reconstruction have made them a popular exploratory tool to detect network-like evolution in genetic datasets. Standard reconstruction methods for circular networks, such as Neighbor-Net, rely on an associated metric on the species set. Such a metric is first estimated from DNA sequences, which leads to a key difficulty: distantly related sequences produce statistically unreliable estimates. This is problematic for Neighbor-Net as it is based on the popular tree reconstruction method Neighbor-Joining, whose sensitivity to distance estimation errors is well established theoretically. In the tree case, more robust reconstruction methods have been developed using the notion of a distorted metric, which captures the dependence of the error in the distance through a radius of accuracy. Here we design the first circular network reconstruction method based on distorted metrics. Our method is computationally efficient. Moreover, the analysis of its radius of accuracy highlights the important role played by the maximum incompatibility, a measure of the extent to which the network differs from a tree.

q-bio.PE↗

arXiv2017

Coalescent-based species tree estimation: a stochastic Farris transform

The reconstruction of a species phylogeny from genomic data faces two significant hurdles: 1) the trees describing the evolution of each individual gene--i.e., the gene trees--may differ from the species phylogeny and 2) the molecular sequences corresponding to each gene often provide limited information about the gene trees themselves. In this paper we consider an approach to species tree reconstruction that addresses both these hurdles. Specifically, we propose an algorithm for phylogeny reconstruction under the multispecies coalescent model with a standard model of site substitution. The multispecies coalescent is commonly used to model gene tree discordance due to incomplete lineage sorting, a well-studied population-genetic effect. In previous work, an information-theoretic trade-off was derived in this context between the number of loci, $m$, needed for an accurate reconstruction and the length of the locus sequences, $k$. It was shown that to reconstruct an internal branch of length $f$, one needs $m$ to be of the order of $1/[f^{2} \sqrt{k}]$. That previous result was obtained under the molecular clock assumption, i.e., under the assumption that mutation rates (as well as population sizes) are constant across the species phylogeny. Here we generalize this result beyond the restrictive molecular clock assumption, and obtain a new reconstruction algorithm that has the same data requirement (up to log factors). Our main contribution is a novel reduction to the molecular clock case under the multispecies coalescent. As a corollary, we also obtain a new identifiability result of independent interest: for any species tree with $n \geq 3$ species, the rooted species tree can be identified from the distribution of its unrooted weighted gene trees even in the absence of a molecular clock.

cs.LG↗

arXiv2016

Phase transition on the convergence rate of parameter estimation under an Ornstein-Uhlenbeck diffusion on a tree

Diffusion processes on trees are commonly used in evolutionary biology to model the joint distribution of continuous traits, such as body mass, across species. Estimating the parameters of such processes from tip values presents challenges because of the intrinsic correlation between the observations produced by the shared evolutionary history, thus violating the standard independence assumption of large-sample theory. For instance Ho and Ané \cite{HoAne13} recently proved that the mean (also known in this context as selection optimum) of an Ornstein-Uhlenbeck process on a tree cannot be estimated consistently from an increasing number of tip observations if the tree height is bounded. Here, using a fruitful connection to the so-called reconstruction problem in probability theory, we study the convergence rate of parameter estimation in the unbounded height case. For the mean of the process, we provide a necessary and sufficient condition for the consistency of the maximum likelihood estimator (MLE) and establish a phase transition on its convergence rate in terms of the growth of the tree. In particular we show that a loss of $\sqrt{n}$-consistency (i.e., the variance of the MLE becomes $Ω(n^{-1})$, where $n$ is the number of tips) occurs when the tree growth is larger than a threshold related to the phase transition of the reconstruction problem. For the covariance parameters, we give a novel, efficient estimation method which achieves $\sqrt{n}$-consistency under natural assumptions on the tree.

q-bio.PE↗

arXiv2014

Likelihood-based tree reconstruction on a concatenation of alignments can be positively misleading

The reconstruction of a species tree from genomic data faces a double hurdle. First, the (gene) tree describing the evolution of each gene may differ from the species tree, for instance, due to incomplete lineage sorting. Second, the aligned genetic sequences at the leaves of each gene tree provide merely an imperfect estimate of the topology of the gene tree. In this note, we demonstrate formally that a basic statistical problem arises if one tries to avoid accounting for these two processes and analyses the genetic data directly via a concatenation approach. More precisely, we show that, under the multi-species coalescent with a standard site substitution model, maximum likelihood estimation on sequence data that has been concatenated across genes and performed under the incorrect assumption that all sites have evolved independently and identically on a fixed tree is a statistically inconsistent estimator of the species tree. Our results provide a formal justification of simulation results described of Kubatko and Degnan (2007) and others, and complements recent theoretical results by DeGorgio and Degnan (2010) and Chifman and Kubtako (2014).

q-bio.PE↗

arXiv2014

Data Requirement for Phylogenetic Inference from Multiple Loci: A New Distance Method

We consider the problem of estimating the evolutionary history of a set of species (phylogeny or species tree) from several genes. It is known that the evolutionary history of individual genes (gene trees) might be topologically distinct from each other and from the underlying species tree, possibly confounding phylogenetic analysis. A further complication in practice is that one has to estimate gene trees from molecular sequences of finite length. We provide the first full data-requirement analysis of a species tree reconstruction method that takes into account estimation errors at the gene level. Under that criterion, we also devise a novel reconstruction algorithm that provably improves over all previous methods in a regime of interest.

q-bio.PE↗

arXiv2013

Alignment-free phylogenetic reconstruction: Sample complexity via a branching process analysis

We present an efficient phylogenetic reconstruction algorithm allowing insertions and deletions which provably achieves a sequence-length requirement (or sample complexity) growing polynomially in the number of taxa. Our algorithm is distance-based, that is, it relies on pairwise sequence comparisons. More importantly, our approach largely bypasses the difficult problem of multiple sequence alignment.

math.PR↗

arXiv2012

Phylogenetic mixtures: Concentration of measure in the large-tree limit

The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practice. Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.

math.PR↗

arXiv2012

An analytical comparison of coalescent-based multilocus methods: The three-taxon case

Incomplete lineage sorting (ILS) is a common source of gene tree incongruence in multilocus analyses. A large number of methods have been developed to infer species trees in the presence of ILS. Here we provide a mathematical analysis of several coalescent-based methods. Our analysis is performed on a three-taxon species tree and assumes that the gene trees are correctly reconstructed along with their branch lengths.

math.PR↗

arXiv2012

Recovering the tree-like trend of evolution despite extensive lateral genetic transfer: A probabilistic analysis

Lateral gene transfer (LGT) is a common mechanism of non-vertical evolution where genetic material is transferred between two more or less distantly related organisms. It is particularly common in bacteria where it contributes to adaptive evolution with important medical implications. In evolutionary studies, LGT has been shown to create widespread discordance between gene trees as genomes become mosaics of gene histories. In particular, the Tree of Life has been questioned as an appropriate representation of bacterial evolutionary history. Nevertheless a common hypothesis is that prokaryotic evolution is primarily tree-like, but that the underlying trend is obscured by LGT. Extensive empirical work has sought to extract a common tree-like signal from conflicting gene trees. Here we give a probabilistic perspective on the problem of recovering the tree-like trend despite LGT. Under a model of randomly distributed LGT, we show that the species phylogeny can be reconstructed even in the presence of surprisingly many (almost linear number of) LGT events per gene tree. Our results, which are optimal up to logarithmic factors, are based on the analysis of a robust, computationally efficient reconstruction method and provides insight into the design of such methods. Finally we show that our results have implications for the discovery of highways of gene sharing.

math.PR↗

arXiv2011

Robust estimation of latent tree graphical models: Inferring hidden states with inexact parameters

Latent tree graphical models are widely used in computational biology, signal and image processing, and network tomography. Here we design a new efficient, estimation procedure for latent tree models, including Gaussian and discrete, reversible models, that significantly improves on previous sample requirement bounds. Our techniques are based on a new hidden state estimator which is robust to inaccuracies in estimated parameters. More precisely, we prove that latent tree models can be estimated with high probability in the so-called Kesten-Stigum regime with $O(log^2 n)$ samples where $n$ is the number of nodes.

math.PR↗

arXiv2011

Phase Transition in Distance-Based Phylogeny Reconstruction

We introduce a new distance-based phylogeny reconstruction technique which provably achieves, at sufficiently short branch lengths, a logarithmic sequence-length requirement---improving significantly over previous polynomial bounds for distance-based methods and matching existing results for general methods. The technique is based on an averaging procedure that implicitly reconstructs ancestral sequences. In the same token, we extend previous results on phase transitions in phylogeny reconstruction to general time-reversible models. More precisely, we show that in the so-called Kesten-Stigum zone (roughly, a region of the parameter space where ancestral sequences are well approximated by "linear combinations" of the observed sequences) sequences of length $O(\log n)$ suffice for reconstruction when branch lengths are discretized. Here $n$ is the number of extant species. Our results challenge, to some extent, the conventional wisdom that estimates of evolutionary distances alone carry significantly less information about phylogenies than full sequence datasets.

math.PR↗

arXiv2011

Identifiability and inference of non-parametric rates-across-sites models on large-scale phylogenies

Mutation rate variation across loci is well known to cause difficulties, notably identifiability issues, in the reconstruction of evolutionary trees from molecular sequences. Here we introduce a new approach for estimating general rates-across-sites models. Our results imply, in particular, that large phylogenies are typically identifiable under rate variation. We also derive sequence-length requirements for high-probability reconstruction. Our main contribution is a novel algorithm that clusters sites according to their mutation rate. Following this site clustering step, standard reconstruction techniques can be used to recover the phylogeny. Our results rely on a basic insight: that, for large trees, certain site statistics experience concentration-of-measure phenomena.

math.PR↗

arXiv2010

On the inference of large phylogenies with long branches: How long is too long?

Recent work has highlighted deep connections between sequence-length requirements for high-probability phylogeny reconstruction and the related problem of the estimation of ancestral sequences. In [Daskalakis et al.'09], building on the work of [Mossel'04], a tight sequence-length requirement was obtained for the CFN model. In particular the required sequence length for high-probability reconstruction was shown to undergo a sharp transition (from $O(\log n)$ to $\hbox{poly}(n)$, where $n$ is the number of leaves) at the "critical" branch length $\critmlq$ (if it exists) of the ancestral reconstruction problem. Here we consider the GTR model. For this model, recent results of [Roch'09] show that the tree can be accurately reconstructed with sequences of length $O(\log(n))$ when the branch lengths are below $\critksq$, known as the Kesten-Stigum (KS) bound. Although for the CFN model $\critmlq = \critksq$, it is known that for the more general GTR models one has $\critmlq \geq \critksq$ with a strict inequality in many cases. Here, we show that this phenomenon also holds for phylogenetic reconstruction by exhibiting a family of symmetric models $Q$ and a phylogenetic reconstruction algorithm which recovers the tree from $O(\log n)$-length sequences for some branch lengths in the range $(\critksq,\critmlq)$. Second we prove that phylogenetic reconstruction under GTR models requires a polynomial sequence-length for branch lengths above $\critmlq$.

math.PR↗